Best Known (74−37, 74, s)-Nets in Base 128
(74−37, 74, 910)-Net over F128 — Constructive and digital
Digital (37, 74, 910)-net over F128, using
- 1281 times duplication [i] based on digital (36, 73, 910)-net over F128, using
- net defined by OOA [i] based on linear OOA(12873, 910, F128, 37, 37) (dual of [(910, 37), 33597, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(12873, 16381, F128, 37) (dual of [16381, 16308, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(12873, 16384, F128, 37) (dual of [16384, 16311, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- discarding factors / shortening the dual code based on linear OA(12873, 16384, F128, 37) (dual of [16384, 16311, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(12873, 16381, F128, 37) (dual of [16381, 16308, 38]-code), using
- net defined by OOA [i] based on linear OOA(12873, 910, F128, 37, 37) (dual of [(910, 37), 33597, 38]-NRT-code), using
(74−37, 74, 4083)-Net over F128 — Digital
Digital (37, 74, 4083)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12874, 4083, F128, 4, 37) (dual of [(4083, 4), 16258, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12874, 4097, F128, 4, 37) (dual of [(4097, 4), 16314, 38]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12874, 16388, F128, 37) (dual of [16388, 16314, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(12874, 16390, F128, 37) (dual of [16390, 16316, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,17]) [i] based on
- linear OA(12873, 16385, F128, 37) (dual of [16385, 16312, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(12869, 16385, F128, 35) (dual of [16385, 16316, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,18]) ⊂ C([0,17]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12874, 16390, F128, 37) (dual of [16390, 16316, 38]-code), using
- OOA 4-folding [i] based on linear OA(12874, 16388, F128, 37) (dual of [16388, 16314, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(12874, 4097, F128, 4, 37) (dual of [(4097, 4), 16314, 38]-NRT-code), using
(74−37, 74, large)-Net in Base 128 — Upper bound on s
There is no (37, 74, large)-net in base 128, because
- 35 times m-reduction [i] would yield (37, 39, large)-net in base 128, but